1. Some types of
POLAR CURVES

2. Sketch these graphs: 𝑦= sin 𝑥 𝑦= cosec 𝑥 𝑦= cos 𝑥 𝑦= sec 𝑥
Polar curves
KUS objectives
BAT sketch curves based on their Polar equations
𝑥=𝑟𝑐𝑜𝑠𝜃
𝑦=𝑟𝑠𝑖𝑛𝜃
𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥
𝑟 2 = 𝑥 2 + 𝑦 2
Starter:
Sketch these graphs:
𝑦= sin 𝑥 𝑦= cosec 𝑥
𝑦= cos 𝑥 𝑦= sec 𝑥
𝑦= tan 𝑥 𝑦= cot 𝑥

3. We ignore situations where r < 0
Notes 1
𝑥=𝑟𝑐𝑜𝑠𝜃
𝑦=𝑟𝑠𝑖𝑛𝜃
𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥
𝑟 2 = 𝑥 2 + 𝑦 2
We do not plot any points for a polar curve that give a negative value of r.
If you think about it, if you get a negative value for r, the logical way to deal with it would be to plot it in the opposite direction
However, changing the direction would mean that the angle used to calculate the value is now different, so the pair of values cannot go together
We ignore situations where r < 0

4. 𝑟=2+ 𝑐𝑜s 𝜃 −𝜋 ≤𝜃≤𝜋 WB12a Cartesian to Polar coordinates I Table
Use Radians
WB12a Cartesian to Polar coordinates I
a) CARDIOID
𝑟=2+ 𝑐𝑜s 𝜃
−𝜋 ≤𝜃≤𝜋
for θ
− 𝜋
− 𝜋
− 𝜋
− 𝜋
− 𝜋 r
1 6 𝜋
1 3 𝜋
1 2 𝜋
2 3 𝜋
5 6 𝜋 𝜋
1.134
1.5 2
2.5
2.866 3
Table
2.866
2.5 2
1.5
1.134 1

5. 𝑟 2 =16 𝑐𝑜s 2𝜃 −𝜋 ≤𝜃≤𝜋 WB12b Cartesian to Polar coordinates I Table
Use Radians
WB12b Cartesian to Polar coordinates I
b) LEMINISCATE
𝑟 2 =16 𝑐𝑜s 2𝜃
−𝜋 ≤𝜃≤𝜋
for θ
− 𝜋
− 𝜋
− 𝜋
− 𝜋
− 𝜋 r
1 6 𝜋
1 3 𝜋
1 2 𝜋
2 3 𝜋
5 6 𝜋 𝜋
Table

6. 𝑟=2 𝑒 𝜃 7 −𝜋 ≤𝜃≤𝜋 WB12c Cartesian to Polar coordinates I Table
Use Radians
WB12c Cartesian to Polar coordinates I
c) SPIRAL
𝑟=2 𝑒 𝜃 7
−𝜋 ≤𝜃≤𝜋
for θ
− 𝜋
− 𝜋
− 𝜋
− 𝜋
− 𝜋 r
1 6 𝜋
1 3 𝜋
1 2 𝜋
2 3 𝜋
5 6 𝜋 𝜋
Table

7. 𝑟=4 sin 3𝜃 −𝜋 ≤𝜃≤𝜋 WB12d Cartesian to Polar coordinates I Table
Use Radians
WB12d Cartesian to Polar coordinates I
d) ROSE
𝑟=4 sin 3𝜃
−𝜋 ≤𝜃≤𝜋
for θ
− 𝜋
− 𝜋
− 𝜋
− 𝜋
− 𝜋 r
1 6 𝜋
1 3 𝜋
1 2 𝜋
2 3 𝜋
5 6 𝜋 𝜋
Table

8. 𝑟=2 𝑐𝑜𝑠 𝜃 𝑟=2 𝑠𝑖𝑛 𝜃 𝑟=2− sin 𝜃 𝑟=2− sin 2𝜃 𝑟=2− sin 3𝜃 Use Radians
WB13 Sketch the curves with these Polar equations
𝑟=2 𝑐𝑜𝑠 𝜃
𝑟=2 𝑠𝑖𝑛 𝜃
𝑟=2− cos 𝜃− 1 4 𝜋
𝑟=2− sin 𝜃
𝑟=2− sin 2𝜃
𝑟=2− sin 3𝜃

10. Skills TASKS: SK103 sketch polar curves SK105 curves match1
𝑥=𝑟𝑐𝑜𝑠𝜃
𝑦=𝑟𝑠𝑖𝑛𝜃
𝜃=𝑎𝑟𝑐𝑡𝑎𝑛 𝑦 𝑥
𝑟 2 = 𝑥 2 + 𝑦 2
Skills TASKS:
SK103 sketch polar curves
SK105 curves match1
SK106 curves match2
SK107 curves to equations
Skills Exercise :
SK104 polar curves

11. Crucial points Summary I (r, θ) origin Initial line
Make sure that you can covert between cartesian and polar coordinates.
cartesian (x, y) polar (r, θ) θ is In the range [-, ] or [0, 2]
2. Remember that cos θ = cos (- θ)
If the equation only involves cos it will be symmetrical in the initial line
3. We can use periodicity of the functions to help us
e.g. tan has a period of . Therefore r = tan 5θ will repeat itself after an angle of

12. One thing to improve is –
KUS objectives BAT sketch curves based on their Polar equations
self-assess
One thing learned is –
One thing to improve is –

13. END